An extended class of scale-invariant and recursive scale space filters

TLDR: The authors show that only a discrete subset of filters gives rise to an evolution which can be characterized by means of a partial differential equation.

Abstract: Explores how the functional form of scale space filters is determined by a number of a priori conditions. In particular, if one assumes scale space filters to be linear, isotropic convolution filters, then two conditions (viz. recursivity and scale-invariance) suffice to narrow down the collection of possible filters to a family that essentially depends on one parameter which determines the qualitative shape of the filter. Gaussian filters correspond to one particular value of this shape-parameter. For other values the filters exhibit a more complicated pattern of excitatory and inhibitory regions. This might well be relevant to the study of the neurophysiology of biological visual systems, for recent research shows the existence of extensive disinhibitory regions outside the periphery of the classical center-surround receptive field of LGN and retinal ganglion cells (in cats). Such regions cannot be accounted for by models based on the second order derivative of the Gaussian. Finally, the authors investigate how this work ties in with another axiomatic approach of scale space operators which focuses on the semigroup properties of the operator family. The authors show that only a discrete subset of filters gives rise to an evolution which can be characterized by means of a partial differential equation. >